Generating Globally Regular Indirect Utility functions

GENERATING GLOBALLY REGULAR INDIRECT UTILITY FUNCTIONS Denis Conni e Economics Depa men , Na ional Uni e si y o I eland Maynoo h Abs ac Despi e hei sca ci y in he li e a u e, an abundance o globally egula indi ec u ili y unc ions, in ol ing as many pa ame e s as desi ed, exis s. They a e easily cons uc ed as a unc ion o simple homo he ic componen u ili ies. JEL Classi ica ion: D 11 Keywo ds: Global egula i y, indi ec u ili y unc ions. Add ess o co espondence: Denis Conni e, Economics Depa men , Na ional Uni e si y o I eland, Maynoo h, Co. Kilda e, I eland. Tel: 353 1 7086299; Fax: 353 1 7083934; e-mail: Denis.Conni [email protected] 2 1. INTRODUCTION Regula i y means ha an indi ec u ili y unc ion complies wi h he cons ain s implied by a ional economic beha iou 1. Regula i y is global i i holds o all (posi i e) p ices and incomes, gi en app op ia e anges o he alues o pa ame e s occu ing in he unc ion. This sho pape is conce ned wi h he gene a ion o such globally egula unc ions. 2. COMBINING HOMOTHETIC COMPONRNTS Conside a se o homo he ic u ili y unc ions, each o he o m (1) ,/),( kk PyyU k  p whe e k  is posi i e and is inc easing in p ices, homogeneous o deg ee k Pk  in p ices and is a conca e unc ion o p ices wi h nega i e de ini e o semi-de ini e Hessian. Then he ecip ocal o is k P con ex in p ices, because 2 2 2 2 32 12 12 i k k i k ki k p P P p P Pp P                 and ji k k j k i k k ji k pp P P p P p P P pp P                          2 23 12 12 , so he Hessian o is 1 k P ' 3 2                    pp kk k PP P, which is posi i e semi-de ini e, minus he Hessian o , which also gi es a nonnega i e de ini e k P ma ix. So is con ex in p, non-dec easing in y, non-inc easing in p and as well as ),( yUkp homogeneous o deg ee ze o in income y and p ices. So i is globally egula . A ew p ope ies o k Ua e wo h examining. As is al eady ob ious, is a conca e unc ion o p ices. Fo posi i e 1 k U 1  , , being an inc easing conca e unc ion o a conca e unc ion, is conca e in     kk UU 1 1 Tha is, a consume maximises di ec u ili y unde a budge cons ain . This implies he indi ec u ili y unc ion should be homogeneous o deg ee ze o in income y and p ices p, non- dec easing in y, non-inc easing in p, and con ex o quasi-con ex in p. These cons ain s imply co esponding condi ions (agg ega ion, homogenei y, Slu sky symme y and nega i i y) on he demand equa ions. ),( yU p 3 p ices. Fo 1  , is an inc easing con ex unc ion o a con ex unc ion and so is con ex in   k U p ices. Fo 1  , i is simple o e i y ha    k Phas a nega i e de ini e o semi-de ini e Hessian and he e o e i s ecip ocal is con ex in p ices and so is     kk UU    k Py k/. So is   k U con ex in p ices o all posi i e  . Also, is con ex in p ices as i equals k Ulog Py kloglog   and since is an inc easing conca e unc ion o a conca e unc ion, i is conca e and minus i is Plog con ex. Now conside he unc ion      1    kkUU , (2) whe e he k  a e non-nega i e and sum o uni y2. Commence wi h posi i e 1   . is conca e   k U in p ices and since sums o conca e unc ions a e conca e,     kkU is conca e. Then     1  kkU is a dec easing unc ion o a quasi-conca e unc ion and so is quasi-con ex. Since             kkkkk UU yy U1 1 1 and  i k k kkkk ip P P UU p U        1 1 1    Uis inc easing in income and dec easing in p ices. I is ob iously homogeneous o deg ee ze o in income and p ices since each is. So U is globally egula o k U1   . Now ake  nega i e and pu    . Then (2) becomes      1  kkUU . As al eady shown is con ex in p ices, so  k U    kkUis con ex and since an inc easing unc ion o a con ex unc ion is quasi-con ex, Uis quasi-con ex in p ices. I is clea ha is again inc easing U in income, dec easing in p ices and homogeneous o deg ee ze o in income and p ices. Finally, o 2 The unc ion (2) is o he o m o he ecip ocal o a CES (cons an elas ici y o subs i u ion) p ice index, al hough he e i is an index o componen u ili ies no p ices. 4 0  , he amilia limi ing a gumen as 0  ( o example, Diewe , 1993) gi es o  k k UU   kk UU loglog  . Since each is con ex in p ices, he sum is, and since an ilog is an inc easing con ex unc ion, k Ulog Uis. So (2) gi es a globally egula u ili y unc ion o all 1   . Fo u ili ies o he o m , which a e special cases o (1), he sums, p oduc s and ha monic k Py/ means, which a e special cases o (2), ha e been examined in Conni e (2002). Howe e , hey a e o limi ed in e es as he esul ing u ili ies mus all be homo he ic. 3. EXAMPLES Conside   jj p y U  1 and   jiij pp y U  2. Bo h a e easily shown o be con ex in p ices, he o me s ic ly so, p o ided he j  and ij  a e posi i e and ij  = ji  . Take 3/1,3/2 2     iand 1   , so ha (2) is a weigh ed ha monic mean o and . I is 1 U2 U 1 2 1 2 1 2 1 23                                    y p y p y pj i ij j j  , (3) Diewe ’s (1974) gene alised Leon ie u ili y unc ion. Diewe p o ed i s global egula i y, bu ha is immedia ely e iden om he de i a ion he e. Mo e impo an ly, (3) is jus one o many possibili ies, all globally egula . Taking 1  , wi h he same  alues as be o e, gi es a weigh ed a i hme ic mean o and 1 U2 U 1 2 1 2 1 1 2 1 3 1 3 2                                              y p y p y pj i ij j j  . (4) The demand sys ems ollowing om (3) and (4) a e no he same and show di e ences ha a e possibly impo an in p ac ice. Fo example, he income elas ici ies o he gene alised Leon ie demand sys em a e 5                                                                2 1 2 1 2 1 2 1 2 1 2 1 2 2 1 y p y p y p y p y p y p w E j i ij j j j i ij i i i i    , which mus be g ea e han a hal . The income elas ici ies co esponding o (4) a e                                                2 1 2 1 2 2 1 2 1 2 1 2 1 2 2 1 2 1 /2 / 2 3 jiijjj j j jiijjj i i i i ppp y p ppp y p w E     ’ which mus be less han 3/2, which migh be mo e accep able han cons aining hem o exceed a hal . Regula i y need no imply he capabili y o model a wide ange o consume beha iou . The gene alised Leon ie ’s de iciencies in his ega d ha e been men ioned by Ca es and Ch is ensen (1980) and Diewe and Wales (1987) and (4) has i s own in lexibili ies. Hoewe e ,  , o a  , could be ea ed as an unknown pa ame e o gain mo e lexibili y. The case o 0  , gi ing p oduc s o u ili ies is pa icula ily in lexible since he esul ing unc ion is s ill homo he ic. Many o he choices o and a e ob iously possible; o example, could be aken o be 1 U2 U1 U j j p y   1, whe e 1    j. Using (2) o combine his wi h he p e ious wi h 2 U  =1 and 2/1 21    gi es 1 2 1 2 1 2                                     y p y p y pj i ij j j   , and he co esponding demand equa ions a e                                                        2 1 2 1 1 2 1 2 1 y p y p y p y p y p y p w j i ij j j ij i j i i j j     . 6 Clea ly, h ee o mo e componen u ili y unc ions could be employed in (2) o p oduce e en mo e hea ily pa ame e ised, bu s ill globally egula , u ili y unc ions. While he lexibili y o he esul ing demand sys ems would be inc eased, especially i  , o he  we e also pa ame e ised, he da a equi emen s o he es ima ion o so many pa ame e s could be a se ious p ac ical di icul y. 4. DISCUSSION This pape has shown how o gene a e highly pa ame e ised globally egula u ili y unc ions om (2) using he globally egula componen s (1). Familia and mo e pa simonious u ili y unc ions could also be seen as ollowing om (2). Hou hakke ’s (1960) indi ec addilog unc ion j j jp y          , whe e i  and i  ha e he same sign and i  >-1, is globally egula and is ob iously (2) applied o j py/ o he powe o j  wi h 1  and jj    . The componen s employed in sec ion 3 could be conside ed as esul ing om applica ion o (2) o s ill simple componen s. Fo example, le kk pyU /, a u ili y unc ion co esponding o expendi u e o he whole budge on good k. Using (2) wi h 1  and jj    gi es he o sec ion 3. 1 Ukiik ppyU /, a u ili y unc ion co esponding o expendi u e o hal he budge on good i and hal on good k, is (2) applied o and i U k Uwi h 0  and 2/1 ki   . Then applying (2) o wi h ik U1   and  o  gi es 2 Uo sec ion 3. O cou se, many u ili y unc ions ha e appea ed in he li e a u e ha a e no o he class gene a ed by (2) om componen s o he o m (1). Bu hey a e no globally egula . REFERENCES Ca es, D. and L. R. Ch is ensen (1980): “Global p ope ies o lexible unc ional o ms”, Ame ican Economic Re iew, 70, 422-432. Conni e, D. (2002): “Sums and P oduc s o U ili y Func ions”, Economic and Social Re iew, 33, 285-295. Diewe , W. E. (1974): “Applica ions o duali y heo y”, in F on ie s o Quan i a i e Economics Vol. II, ed. By M. D In iliga o , and D. A.Kend ick, Ams e dam: No h-Holland, 106-171. Diewe , W. E.and T. J. Wales (1987): “Flexible unc ional o ms and global cu a u e Condi ions”, Econome ica, 55, 43-68. Hou hakke , H. S. 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